Fracture paths described by generalized random walks on curved spatiotemporal surfaces

Abstract

An expression representing trajectories of fracture paths is obtained as a first arrival length. The first arrival length is a function representing a quantity generalized from the first passage time. The expression is the average value of the first arrival length found by solving a backward Fokker–Planck equation on a curved spatiotemporal surface. The backward FP equation is derived from generalized random walks (GRW). In the GRW, jump probabilities are specified by additional arbitrary functions. Furthermore, we propose a stochastic differential equation of random variables representing the fracture paths.